The Khovanov homology of knots

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

The Khovanov homology of knots

<p>The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.</p>

Kauffman bracket skein module of the connected sum of handlebodies and non-injectivity

For the handlebody [Formula: see text] of genus [Formula: see text], Przytycki studied the (Kauffman bracket) skein module [Formula: see text] of the connected sum [Formula: see text] at [Formula: see text]. One of his results is that, in the case when [Formula: see text] is invertible for any [Formula: see text], a homomorphism [Formula: see text] is an isomorphism, which is induced by a natural way. In this paper, in the case when [Formula: see text], the ground ring is [Formula: see text], and [Formula: see text] is a [Formula: see text]-th root of unity ([Formula: see text]), we show that [Formula: see text] is not injective.

Classification of prime links in the thickened torus having crossing number 5

The goal of this paper is to tabulate all prime links in the thickened torus [Formula: see text] having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus [Formula: see text] having exactly 5 crossings. To this end, we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus [Formula: see text] in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus [Formula: see text]. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.

Picture-valued parity-biquandle bracket

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.